A Lesson On Pi
What do you think of when you hear Pi? What does Pi really mean? Many adults remember learning about Pi in math classes, but many don’t really understand what it means. Many people will think of 3.14, and some may even remember the fraction 22/7. Many of us were taught the formula 2Pr or Pr(squared) (P represents Pi), and we might remember it has something to do with circumference or diameter. If you can remember that C= 2Pr and A= Pr(squared), you were probably a pretty good math student. You were probably taught the formulas and how to use them, and probably have experience in finding the area and circumference of circles when you are given the diameter or the radius. Assignments often included doing a whole page of these problems with the formulas written at the top of the page for reference. If we forgot how to do these problems, it was easy to find a “cloned” example, or maybe the teacher would show us how to do one so that we could do the rest. Being able to complete “clone” problems would hold us over until the test, and then, most likely, we didn’t need to use that “knowledge” until the next year, when, once again, the teacher would show us an example and we would repeat the similar process. Years ago, that was the typical math experience. In some classrooms it is still happening today.
In the last thirty years a great deal of research has been done on how kids learn and understand math. Getting them to figure it out for themselves, through understanding, is the goal. Here is an example of how the concept of Pi can be learned through teaching for understanding.
Students are asked to measure 10-15 various circular objects in class. (These can easily be found in most “Tupperware” drawers.) Students then measure the diameter and circumference using a tape measure, and then they put the measurements into a data table.
Object
Circumference
Diameter
C÷D
Lid A
35.3
11
3.21
Lid B
124.3
41
3.03
Lid C
76.1
25.2
3.01
Next, they graph pairs and look for a pattern. Most students easily find the linear pattern. Then they are able to make predictions of a circle’s circumference when they are given the diameter. Kids discover that the circumference is always about three times bigger than the diameter. The idea that the circumference is three times greater than the diameter is a very simple concept when discovered in this way. This is when the number for Pi (3.14) is introduced. The circumference is not just three times longer than the radius; it is exactly 3.14, or Pi, times longer. This simple activity helps kids understand the concept of Pi, as opposed to just doing the calculations to figure out the area and circumference of circles.
Hopefully you’ll agree that the above activity is more conducive to understanding the concept of Pi. Plugging numbers into a formula is not a bad skill to learn, but the concepts remain abstract. The above activity is visual, reaching those with another learning style, and it helps students discover that a circle’s diameter is about one third of its circumference. Kids soon begin to predict, well before the last lid is measured, that the circumference will be approximately three times larger than the diameter. It is at this point when they will be asked to make predictions about the circumference of other circles when they have just been given the diameter. In measuring circles, it is much easier to measure the diameter than the circumference. By the end of the lesson, students realize that it is not necessary to measure both.
Teaching for understanding can be frustrating as a teacher, but it can also be incredibly rewarding.
Hopefully this explanation of teaching the concept of Pi has helped you to have a clearer understanding about teaching and understanding math.
What do you think of when you hear Pi? What does Pi really mean? Many adults remember learning about Pi in math classes, but many don’t really understand what it means. Many people will think of 3.14, and some may even remember the fraction 22/7. Many of us were taught the formula 2Pr or Pr(squared) (P represents Pi), and we might remember it has something to do with circumference or diameter. If you can remember that C= 2Pr and A= Pr(squared), you were probably a pretty good math student. You were probably taught the formulas and how to use them, and probably have experience in finding the area and circumference of circles when you are given the diameter or the radius. Assignments often included doing a whole page of these problems with the formulas written at the top of the page for reference. If we forgot how to do these problems, it was easy to find a “cloned” example, or maybe the teacher would show us how to do one so that we could do the rest. Being able to complete “clone” problems would hold us over until the test, and then, most likely, we didn’t need to use that “knowledge” until the next year, when, once again, the teacher would show us an example and we would repeat the similar process. Years ago, that was the typical math experience. In some classrooms it is still happening today.
In the last thirty years a great deal of research has been done on how kids learn and understand math. Getting them to figure it out for themselves, through understanding, is the goal. Here is an example of how the concept of Pi can be learned through teaching for understanding.
Students are asked to measure 10-15 various circular objects in class. (These can easily be found in most “Tupperware” drawers.) Students then measure the diameter and circumference using a tape measure, and then they put the measurements into a data table.
Next, they graph pairs and look for a pattern. Most students easily find the linear pattern. Then they are able to make predictions of a circle’s circumference when they are given the diameter. Kids discover that the circumference is always about three times bigger than the diameter. The idea that the circumference is three times greater than the diameter is a very simple concept when discovered in this way. This is when the number for Pi (3.14) is introduced. The circumference is not just three times longer than the radius; it is exactly 3.14, or Pi, times longer. This simple activity helps kids understand the concept of Pi, as opposed to just doing the calculations to figure out the area and circumference of circles.
Hopefully you’ll agree that the above activity is more conducive to understanding the concept of Pi. Plugging numbers into a formula is not a bad skill to learn, but the concepts remain abstract. The above activity is visual, reaching those with another learning style, and it helps students discover that a circle’s diameter is about one third of its circumference. Kids soon begin to predict, well before the last lid is measured, that the circumference will be approximately three times larger than the diameter. It is at this point when they will be asked to make predictions about the circumference of other circles when they have just been given the diameter. In measuring circles, it is much easier to measure the diameter than the circumference. By the end of the lesson, students realize that it is not necessary to measure both.
Teaching for understanding can be frustrating as a teacher, but it can also be incredibly rewarding.
Hopefully this explanation of teaching the concept of Pi has helped you to have a clearer understanding about teaching and understanding math.
Mr. Peterson